expax (section 3.5)

Average Error: 29.5 → 0.3

Time: 27.5s

Precision: 64

Math

\[e^{a \cdot x} - 1\]

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.0001598576098310202:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x}} \cdot \sqrt[3]{e^{a \cdot x}}\right) \cdot \sqrt[3]{e^{a \cdot x}} - 1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot x + \left(\left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) \cdot \frac{1}{6}\right) + \left(a \cdot x\right) \cdot \left(\frac{1}{2} \cdot \left(a \cdot x\right)\right)\\ \end{array}\]

C

double f(double a, double x) {
        double r5082501 = a;
        double r5082502 = x;
        double r5082503 = r5082501 * r5082502;
        double r5082504 = exp(r5082503);
        double r5082505 = 1.0;
        double r5082506 = r5082504 - r5082505;
        return r5082506;
}

↓

double f(double a, double x) {
        double r5082507 = a;
        double r5082508 = x;
        double r5082509 = r5082507 * r5082508;
        double r5082510 = -0.0001598576098310202;
        bool r5082511 = r5082509 <= r5082510;
        double r5082512 = exp(r5082509);
        double r5082513 = cbrt(r5082512);
        double r5082514 = r5082513 * r5082513;
        double r5082515 = r5082514 * r5082513;
        double r5082516 = 1.0;
        double r5082517 = r5082515 - r5082516;
        double r5082518 = r5082509 * r5082509;
        double r5082519 = r5082509 * r5082518;
        double r5082520 = 0.16666666666666666;
        double r5082521 = r5082519 * r5082520;
        double r5082522 = r5082509 + r5082521;
        double r5082523 = 0.5;
        double r5082524 = r5082523 * r5082509;
        double r5082525 = r5082509 * r5082524;
        double r5082526 = r5082522 + r5082525;
        double r5082527 = r5082511 ? r5082517 : r5082526;
        return r5082527;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.5
Target	0.2
Herbie	0.3

\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (* a x) < -0.0001598576098310202

   1. Initial program 0.0

      \[e^{a \cdot x} - 1\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 0.1

      \[\leadsto \color{blue}{\left(\sqrt[3]{e^{a \cdot x}} \cdot \sqrt[3]{e^{a \cdot x}}\right) \cdot \sqrt[3]{e^{a \cdot x}}} - 1\]

   if -0.0001598576098310202 < (* a x)

   1. Initial program 44.2

      \[e^{a \cdot x} - 1\]

   2. Taylor expanded around 0 14.2

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(a \cdot x + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\right)}\]

   3. Simplified 0.5

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right) + \left(a \cdot x + \left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right)\right) \cdot \frac{1}{6}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.0001598576098310202:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x}} \cdot \sqrt[3]{e^{a \cdot x}}\right) \cdot \sqrt[3]{e^{a \cdot x}} - 1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot x + \left(\left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) \cdot \frac{1}{6}\right) + \left(a \cdot x\right) \cdot \left(\frac{1}{2} \cdot \left(a \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 
(FPCore (a x)
  :name "expax (section 3.5)"
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 1/10) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

  (- (exp (* a x)) 1))